a r X i v : 1 2 0 5 . 5 1 9 5 v 2 [ m a t h  p h ] 1 1 O c t 2 0 1 2
A new description of space and time using Cliﬀordmultivectors
James M. Chappell
†
, Nicolangelo Iannella
†
, Azhar Iqbal
†
, Mark Chappell
‡
, DerekAbbott
†
†
School of Electrical and Electronic Engineering, University of Adelaide, South Australia 5005,Australia
‡
Griﬃth Institute, Griﬃth University, Queensland 4122, Australia
Abstract
Following the development of the special theory of relativity in 1905, Minkowski proposed a uniﬁed space and time structure consisting of three space dimensions and onetime dimension, with relativistic eﬀects then being natural consequences of this spacetime geometry. In this paper, we illustrate how Cliﬀord’s geometric algebra that utilizesmultivectors to represent spacetime, provides an elegant mathematical framework for thestudy of relativistic phenomena. We show, with several examples, how the application of geometric algebra leads to the correct relativistic description of the physical phenomenabeing considered. This approach not only provides a compact mathematical representation to tackle such phenomena, but also suggests some novel insights into the nature of time.
Keywords:
Geometric algebra, Cliﬀord space, Spacetime, Multivectors, Algebraicframework
1. Introduction
The physical world, based on early investigations, was deemed to possess three independent freedoms of translation, referred to as the three dimensions of space. This naiveconclusion is also supported by more sophisticated analysis such as the existence of onlyﬁve regular polyhedra and the inverse square force laws. If we lived in a world with fourspatial dimensions, for example, we would be able to construct six regular solids, and inﬁve dimensions and above we would ﬁnd only three [1]. Gravity and the electromagneticforce laws have also been experimentally veriﬁed to follow an inverse square law to veryhigh precision [2], indicating the absence of additional macroscopic dimensions beyondthree space dimensions. Additionally Ehrenfest [3] showed that planetary orbits are notstable for more than three space dimensions, with Tangherlini [4] extending this resultfor electronic orbitals around atoms. The importance of the inverse square force law canbe seen from Bertrand’s theorem from classical mechanics, which states that ‘The only
Email address:
james.m.chappell@adelaide.edu.au
(James M. Chappell
†
)
Preprint submitted to Physics Letters A October 12, 2012
central forces that result in closed orbits for all bound particles are the inversesquarelaw and Hooke’s law.’ [5].The concept of time is typically modeled as a fourth Euclideantype dimension appended to the three dimensions of spatial movement forming Minkowski spacetime [6]with spacetime events modeled as fourvectors. Alternative algebraic formalisms thoughhave been proposed such as STA (Space Time Algebra) [7, 8, 9, 10, 11], or APS (Algebraof Physical Space) [12]. The representation of time within the Minkowski framework andalso these alternate approaches interpret time as a Cartesiantype dimension. Howeverwith the observed nonCartesian like behavior of time, such as the time axis possessinga negative contribution to the Pythagorean distance, and the observed inability to freelymove within the time dimension as is possible with space dimensions suggests that analternate representation might be preferable. The results of string theory also suggestthat the ordinary formulation of physics, in a spacetime with three space dimensionsand one time dimension, is insuﬃcient to describe our world [13, 14, 15]. Also, Tiﬀtexplained the apparent quantization of the cosmological redshift [16, 17] with a modelusing time with three dimensions [18, 19, 20, 21], additionally the spinorial nature of time noted by another author [22].Chappell et al. [23] recently employed a twodimensional multivector as an algebraicmodel for spacetime in the plane, with time represented as a bivector. A bivector ingeometric algebra represents an oriented unit area, which acts as a rotation operatorfor the plane. Now, through generalizing this twodimensional multivector descriptionto threedimensions we produce a three dimensional vector
x
and time becomes a unitarea oriented within three dimensions, represented as
ic
t
, which can be combined intoa multivector
x
+
ic
t
. This now gives rise to symmetry between the space and timecoordinates, both being represented as threevectors in our framework, while still producing the results of special relativity. In fact this duality between space and time, withboth possessing the same degrees of freedom, is only possible in three dimensions—intwo dimensions for example we have two degrees of freedom for translation but only onefor rotation [24, 25].Cliﬀord’s geometric algebra was ﬁrst published in 1873, extending the work of Grassman and Hamilton, creating a single uniﬁed real mathematical framework over Cartesian coordinates, which naturally included the algebraic properties of scalars, complexnumbers, quaternions and vectors into a single entity, called the multivector [26]. Weﬁnd that this general algebraic entity, as part of a real threedimensional Cliﬀord algebra
Cl
3
,
0
(
ℜ
), provides an equivalent representation to a Minkowski vector space
ℜ
3
,
1
[27, 28],with signiﬁcant beneﬁts in assisting intuition and in providing an eﬃcient representation.In order to represent the three independent dimensions of space, Cliﬀord deﬁned threealgebraic elements
e
1
,
e
2
and
e
3
, with the expected unit vector property
e
21
=
e
22
=
e
23
= 1 (1)but with each element anticommuting, that is
e
i
e
j
=
−
e
j
e
i
, for
i
=
j
. We then ﬁnd thatthe composite algebraic element, the trivector
i
=
e
1
e
2
e
3
squares to minus one, that is,
i
2
= (
e
1
e
2
e
3
)
2
=
e
1
e
2
e
3
e
1
e
2
e
3
=
−
e
1
e
1
e
2
e
2
=
−
1, assuming an associative algebra, andas it is found to commute with all other elements it can be used interchangeably with theunit imaginary i =
√ −
1. We continue to follow this notational convention throughoutthe paper, denoting the Cliﬀord trivector with
i
, the scalar imaginary i =
√ −
1 and the2
bivector of the plane with an iota,
ι
=
e
1
e
2
. A general Cliﬀord multivector for threespacecan be written by combining all available algebraic elements
a
+
x
1
e
1
+
x
2
e
2
+
x
3
e
3
+
i
(
t
1
e
1
+
t
2
e
2
+
t
3
e
3
) +
ib,
(2)where
a
,
b
,
x
k
and
t
k
are real scalars, and
i
is the trivector. We thus have eight degreesof freedom present, in which we use
x
=
x
1
e
1
+
x
2
e
2
+
x
3
e
3
to represent a Cartesiantypevector, and the bivector
i
(
t
1
e
1
+
t
2
e
2
+
t
3
e
3
) =
t
1
e
2
e
3
+
t
2
e
3
e
1
+
t
3
e
1
e
2
to representtime, forming a time vector
t
=
t
1
e
1
+
t
2
e
2
+
t
3
e
3
. Thus in our framework the temporalcomponents are described by the three bivector components of the multivector. However,when we move into the rest frame of the particle, we only require a single bivectorcomponent, so that time becomes one dimensional as conventionally assumed. That is
ℜ
3
is the exterior algebra of
ℜ
3
which produces the space of multivectors
ℜ⊕ℜ
3
⊕
2
ℜ
3
⊕
3
ℜ
3
, an eightdimensional real vector space denoted by
Cl
3
,
0
(
ℜ
), with thecomplex numbers and quaternions as subalgebras. The presence of the quaternions as asubalgebra, represented as
a
+
i
t
, is signiﬁcant as Hamilton showed that they providethe correct algebra for rotations in three dimensions. The complex numbers using thedescription in Eq. (2) would be described as
a
+
ib
or by using one of the bivectors, suchas
a
+
ie
k
.It might be argued that four dimensional spacetime is required to describe the Diracequation, for example, typically described using the four dimensional Dirac matrices as
γ
µ
∂
µ

ψ
=
−
i
m

ψ
, where i =
√ −
1 and

ψ
is an eight dimensional spinor. Howeverit has been shown [29, 30] that it can be reduced to an equivalent equation in threedimensional space using
Cl
3
,
0
(
ℜ
), being written as
∂ψ
=
−
mψ
∗
ie
3
,
(3)with the multivector gradient operator
∂
=
∂
t
+
∇
, with
∇
being the threegradient, and
ψ
the multivector of three dimensions, shown in Eq. (2), and the operation
ψ
∗
ﬂippingthe sign of the vector and trivector components. This can also be compared with thewell known form of Maxwell’s equations in threespace
∂ψ
=
J
, with the electromagneticﬁeld represented by the multivector
ψ
=
E
+
i
B
and sources
J
=
ρ
−
J
. Hence themultivector of
Cl
3
,
0
(
ℜ
) uniﬁes complex numbers, spinors, fourvectors and tensors intoa single algebraic object. We now utilize the basic properties of the Cliﬀord multivectorto produce an alternate algebraic representation of space and time.
Cliﬀord multivector spacetime
It was shown previously [23] that spacetime events can be described with the multivector of two dimensions
X
=
x
+
ιct,
(4)having the appropriate properties to reproduce the results of special relativity, assuminginteractions are planar, with
x
=
x
1
e
1
+
x
2
e
2
representing a vector in the plane and
t
the observer time, and
ι
=
e
1
e
2
the bivector of the plane. Squaring the coordinatemultivector we ﬁnd
X
2
=
x
2
−
c
2
t
2
, thus producing the correct spacetime distance. Thisrepresentation of coordinates, along with an electromagnetic ﬁeld
F
=
E
+
ιcB
, are sub ject to a Lorentz transformation of the form
L
= e
φ
ˆ
v
/
2
e
ιθ/
2
, with the ﬁrst term deﬁning3
boosts and the second term rotations, where the transformed coordinates or transformedﬁeld are given by
X
′
=
LXL
†
, where
L
†
= e
−
ιθ/
2
e
−
φ
ˆ
v
/
2
. That is, a combined boost anda rotation can be written as
X
′
= e
φ
ˆ
v
/
2
e
ιθ/
2
X
e
−
ιθ/
2
e
−
φ
ˆ
v
/
2
.
(5)We can see that the multivector in Eq. (4) essentially sets up two orthogonal directions
x
and
ι
=
ie
3
, and so we can generalize this structure to three dimensions through a generalrotation of the
e
1
e
2
plane into threespace forming the threedimensional multivector
X
=
x
+
ic
t
,
(6)where
x
=
x
1
e
1
+
x
2
e
2
+
x
3
e
3
is the coordinate vector and
t
=
t
1
e
1
+
t
2
e
2
+
t
3
e
3
atime vector, with
i
=
e
1
e
2
e
3
the trivector. To maintain the orthogonality implied by themultivector in two dimensions we therefore have the constraints
x
·
t
= 0. Now, squaringthe multivector we ﬁnd
X
2
= (
x
+
ic
t
)(
x
+
ic
t
) (7)=
x
2
−
c
2
t
2
+ 2
ic
x
·
t
=
x
2
−
c
2
t
2
using the fact that
xt
+
tx
= 2
x
·
t
= 0, referring to Eq. (A.3) in the Appendix, thusproducing the correct spacetime distance in three dimensions. Thus an event can bedescribed by a position vector in three dimensions, and a three dimensional time vectorin our framework, and to maintain Lorentz invariance we require these two vectors tobe orthogonal. Hence Einstein’s assertion of the invariant distance in Eq. (7) in histheory of special relativity, from this new viewpoint, becomes simply an assertion of theorthogonality of the space and time threevectors describing an event in Eq. (6).We have from Eq. (4) the multivector diﬀerential
dX
=
d
x
+
icd
t
.
(8)For the rest frame of the particle we have
dX
20
=
−
c
2
dτ
2
, which deﬁnes the propertime
τ
of the particle. In the comoving frame of the particle the time vector does notrequire three components with which to describe its orientation, but only has one degreeof freedom deﬁning its length. We have assumed that the speed of light
c
is the samein the rest and the moving frame, as required by Einstein’s second postulate. Now, if the spacetime distance deﬁned in Eq. (7) is invariant under the Lorentz transformationsdeﬁned later in Eq. (16), then we can equate the rest frame interval to the moving frameinterval, giving
c
2
dτ
2
=
c
2
d
t
2
−
d
x
2
=
c
2
d
t
2
−
v
2
d
t
2
=
c
2
d
t
2
1
−
v
2
c
2
,
(9)assuming
d
x
2
=
v
2
d
t
2
, a vector equation that we conﬁrm shortly. Taking the squareroot of Eq. (9), we ﬁnd the time dilation formula

d
t

=
γdτ
where
γ
= 1
1
−
v
2
/c
2
.
(10)4
This equation implies that the length of the time vector

d
t

, in this formalism, equatesto the scalar time variable
t
typically employed in special relativity. From Eq. (8), wecan now calculate the proper velocity through diﬀerentiating with respect to the scalarproper time, giving a multivector representing velocity
U
=
dX dτ
=
d
x

d
t

d
t

dτ
+
icd
t
dτ
=
γ
(
v
+
i
c
)
,
(11)where we use

d
t

dτ
=
γ
and
v
=
d
x

d
t

and the speed of light now becomes vectorial in thedirection of the time vector
c
=
c
ˆ
t
. We can then ﬁnd
U
2
=
γ
2
(
v
+
i
c
)
2
=
1
−
v
2
c
2
−
1
(
v
2
−
c
2
+ 2
i
v
·
c
) =
−
c
2
,
(12)with the orthogonality of space and time
x
·
t
= 0, implying
v
·
c
= 0. We deﬁne themomentum multivector
P
=
mU
=
γm
v
+
iγm
c
=
p
+
i
E
c ,
(13)with the relativistic momentum
p
=
γm
v
and the total energy now having a vectorialnature
E
=
γmc
c
.Now, as
U
2
=
−
c
2
, then
P
2
=
−
m
2
c
2
is an invariant describing the conservation of momentum and energy, which gives
P
2
c
2
=
p
2
c
2
−
E
2
=
−
m
2
c
4
,
(14)or
E
2
=
m
2
c
4
+
p
2
c
2
, the relativistic expression for the conservation of momentumenergy, using the orthogonality of momentum and energy from
p
·
E
=
γ
2
m
2
c
(
v
·
c
) = 0.The vectorial nature of energy, analogous to the Poynting vector describing energy ﬂow,is possible because for a Cliﬀord vector
E
, we ﬁnd that
E
2
is in general a scalar quantityand so can satisfy the Einstein energymomentum relation as shown. At rest the energy
E
0
=
mc
c
has a vectorial nature and hence the Einstein momentumenergy relationin Eq. (14) can be naturally interpreted as a Pythagorean triangle relation between aconstant rest energy vector
mc
c
and the momentum vector
p
c
.
1.1. The Lorentz Group
The Lorentz transformations describe the transformations for observations betweeninertial systems in relative motion. The set of transformations describing rotationsand boosts connected with the identity is referred to as the restricted Lorentz group
SO
+
(3
,
1). We ﬁnd that the exponential of the bivector e
i
ˆ
u
θ
, describes rotations in theplane
i
ˆ
u
, as shown in Eq. (B.3), however, more generally, we can deﬁne the exponentialof a full multivector
M
deﬁned as in Eq. (2), by constructing the Taylor seriese
M
= 1 +
M
+
M
2
2! +
M
3
3! +
...,
(15)which is absolutely convergent for all multivectors
M
[31]. We ﬁnd that the exponentialof a pure vector e
φ
ˆ
v
describes boosts, and so if we deﬁne the combined operator consistingof a boost and rotation
L
= e
φ
ˆ
v
e
i
ˆ
w
θ
,
(16)5